Thinking Inside of the Box

Illustration of the expansion of the Universe ...
Illustration of the expansion of the Universe after the Big bang. In Bulgarian. (Photo credit: Wikipedia)

Science aims to explain things, and by extension to explain everything. Is this even possible? Suppose the Universe consisted of a box, 20 million metres in each direction. Scientists inside this box could investigate this universe, but could they explain everything about this universal Box?

Suppose that the Box had impenetrable walls, so scientists could not probe outside of it. So they could say that the width, height, depth of the universe was 20 million metres and they could describe what was in it. They could also say that one side of the cube attracted everything in the Box and that side could be labelled “down” and the opposite side “up”.

English: Snapshot from a simulation of large s...
English: Snapshot from a simulation of large scale structure formation in a ΛCDM universe. The size of the box is (50 h -1 Mpc) 3 . Run using GADGET (GPL software) (Photo credit: Wikipedia)

There also might be statistical laws, so that the temperature, on average, might be 20 degrees Celsius, but could differ from that norm from place to place and from time to time. Box scientists might determine that everything appeared to be made up of tiny indivisible particles. Box atoms.

Some Box philosophers might ponder what was beyond the limits of the Box. They’d ponder the fact that starting from one side of the Box, one could travel 20 million metres in a perpendicular direction, but one could not travel 20 million and one metres. Why not?


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I’m sure that they would have plenty of theories. For instance, one philosopher might contend that the Box was embedded in an infinite impenetrable bedrock, while another might say that it was obvious – the Box was embedded in nothing. No space, no time, no thing!

Meanwhile scientists probing the Box atoms might split them and discover a whole new world of sub-atomic particles. Others might conceive of space in the Box as being a seething mass of pairs of virtual particles, being created and moving apart for a brief instant and then merging into nothing, no thing, again.

English: Tracks of ionizing radiation in a clo...
English: Tracks of ionizing radiation in a cloud chamber (thick, short: alpha particles; long, thin: beta particles). Français : Traces d’ionisation matérialisées sous forme de micro-trainées de condensation par des particules radioactives dans une chambre à brouillard ; Les trainées épaisses et courtes signalent des particules alpha ; les longues et files matérialisent le passage de particules beta). (Photo credit: Wikipedia)

But, says one bright spark, what about a particle pair created on the boundary of the Box? One particle would enter the Box, and the other would travel somewhere else! This would lead to other speculation – if the second particle travelled in another Box, then that other Box would presumably be a mirror image of our Box!

Such speculation would wait on experimentation by the Box scientists and I’m aware that I cannot push the Box analogy too far with out it breaking. But, just as in the case of the Box scientists, philosophers and scientists in this Universe have similar issue.

An illustration of a ramified analogy, one com...
An illustration of a ramified analogy, one component of Gordon Pask’s Conversation Theory. Self-made (Photo credit: Wikipedia)

In our Universe there are no bounds (under current theories, I believe) but that doesn’t mean that we can’t speculate about what is beyond our Universe, whatever “beyond” may mean in this context.

The Box scientists could potentially explain every thing in the Box, maybe even the fact that it had existed, pretty much unchanged (on average) for all time, and that is periodically, over astronomically long time scale is doomed to repeat itself, time and time again.

Mesquita, repeat ad infinitum
Mesquita, repeat ad infinitum (Photo credit: Wikipedia)

When they go further than that, it is pure speculation, as all the data that they have relates to the Box. They have no data from outside of the Box. All the waves and particles that are observed originate in the Box. All the forces and fields are part of the Box. While scientists may speculate about “other Boxes”, that is all that they can do.

That’s the problem. The Box scientists, and the scientists from our Universe, can only observe events in the Universe in which they are embedded. Observations relate only to events in the local Universe.

English: Multiverse, a light sculpture by Leo ...
English: Multiverse, a light sculpture by Leo Villareal featuring 41,000 computer-programmed LED nodes, located between the National Gallery of Art’s East and West Buildings, on the National Mall in Washington, D.C. (Photo credit: Wikipedia)

Some conjectures suggest that our Universe is one of many universes all linked together in some way. Some conjectures suggest that the laws of our Universe apply in many other similar universes separate from ours. Some people conjecture that universes may exist where there are no laws or the laws that there are have no similarity in any way to the laws of our Universe.

In the Box universe these conjecture would amount to ideas that there may be other Box universes out there with similar laws to the Box universe, maybe linked in some way to the hypothetical Box universe. There may even be universes which have laws which are not at all similar to those of the Box universe. For instance a universe which springs from a single point in a vast explosion and expands at a vast rate either forever or to a certain point only to collapse once again. How bizarre!

The Big Bang era of the universe, presented as...
The Big Bang era of the universe, presented as a manifold in two dimensions (1-space and time); the shape is right (approximately), but it’s not to scale. (Photo credit: Wikipedia)

The Box scientists would not have any way to decide whether or not their were any other Boxes as their observations would only observe events in their own Box. The only way that events in one Box could possibly affect the events in another Box would be if there was a link between them in some way.

This doesn’t necessarily mean that the event would be observable as the effect of one universe on the other universe. It would just appear as an event in each universe as it transpires as a result of the laws of the universe in question.


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The theory may posit a link between two universes but the events in one universe can only result from events within that universe. If this were not so, the event in the universe would appear to happen without any causation in the universe. In other words it would be an anomaly or a miracle.

In other words, suppose a scientist in one universe knows of a law where he can cause an effect in another universe. If he can cause this effect in his universe then in the other universe something will also appear to cause this effect. Maybe this cause will be a scientist in the other universe trying to create an effect in the first universe!


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This possible symmetry of cause and effect across more than one universe would mean that it would be difficult if not impossible to detect the presence of another universe by its effects on our universe.

The person in the Box universe would likely be in the same position. This means that he would never know if there were anything outside of his 20 million metre cube. He could postulate an infinite series of Boxes stacked like bricks in an endless array. Or he could postulate Boxes grouped into “houses”. Or he could postulate that his was the only Box and that speculations about universes started from “Big Bang” explosions are mere fiction.

Detail of the bricks in the Great Wall at Muti...
Detail of the bricks in the Great Wall at Mutianyu. (Photo credit: Wikipedia)

Fractals

A Julia set, a fractal related to the Mandelbr...
A Julia set, a fractal related to the Mandelbrot set (Photo credit: Wikipedia)

Now and then I fire up one of those programs that displays a fractal on the screen. These programs use mathematical programs to display patterns on the screen. Basically the program picks the coordinates of a pixel on the screen and feeds the resulting numbers to the program. Out pop two more numbers. These are fed back to the program and the process is repeated.

There are three possible outcomes from this process.

Firstly, the situation could be reached where the numbers being input to the program also pop out of the program. Once this situation is reached it is said that the program has converged.

Convergent light beam passing through a square...
Convergent light beam passing through a square hole (Photo credit: Wikipedia)

Secondly, the numbers coming out of the program can increase rapidly and without bounds. the program can be said to be diverging.

Thirdly, the results of the calculation could meander around without ever diverging or converging.

English: The Markov chain for the drunkard's w...
English: The Markov chain for the drunkard’s walk (a type of random walk) on the real line starting at 0 with a range of two in both directions. (Photo credit: Wikipedia)

A point where the program converges can then be coloured white. Where it diverges, the point or pixel can be coloured black. A point where the program seems to neither converge nor diverge can then be coloured grey. A pattern will then appear in the three colours which is defined by the equation used.

Anyone who has seen fractals and fractal programs will realise that a three colour fractal is pretty boring as compared to other published fractal images. Indeed the process that I have described is pretty basic. A better image could be drawn by colouring points differently depending on how fast the program converges to a limit. This obviously requires a definition of what constitutes convergence to a limit.

Fractal Art
Fractal Art (Photo credit: Wikipedia)

Convergence is a tricky concept which I’m not going to go into, but to compute it to say in a computer program you have to take into account the errors and rounding introduced by the way that a computer works. In particular the computer has a largest number which it can physically hold, and a smallest number. Various mathematical techniques can be used to extend this, but the extra processing required means that the program slows down.

[Fractal]
[Fractal] (Photo credit: Wikipedia)
I’m not going to explain how this difficulty is circumvented, since I don’t know! However the fact is that the computer generated fractals are fascinating. Most will allow you to continually zoom in on a small area, revealing fantastic “landscapes” which demonstrate similar features at all the descending levels. Similar, but not the same.

fractal landscape
fractal landscape (Photo credit: Wikipedia)

The above far from rigorous description describes one type of fractal of which there are various sorts. Others are described on the Wikipedia page on the subject.

Another interesting fractal is created on the number line. Take a fixed part of the number line, say from 0 to 1, and divide it into three parts. Rub out the middle one third. This leaves two smaller lines, from 0 to 1/3 and from 2/3 to 1. Divide these lines into three parts and perform the same process. Soon, all that is left is practically nothing. This residue is known as the Cantor set, after the mathematician Georg Cantor.

English: A Cantor set Deutsch: Eine Cantor-Men...
English: A Cantor set Deutsch: Eine Cantor-Menge Svenska: Cantordamm i sju iterationer, en fraktal (Photo credit: Wikipedia)

This particular fractal can be generalised to two, three, or even higher dimensions. The two dimensional version is called the Sierpinski curve and the three dimensional version is called the Menger sponge.

One of the fractal curves that I was interested in was the Feigenbaum function. This fractal shows a “period doubling cascade” as shown in the first diagram in the above link. If you see some versions of this diagram the doubling points (from which the constant is determined) often look sharply defined.

English: A very old ficus tree in São Paulo, B...
English: A very old ficus tree in São Paulo, Brasil. Deutsch: Ein sehr alter Feigenbaum in São Paulo, Brasilien. Português do Brasil: Uma figueira muito antiga nas ruas de São Paulo, Brasil. (Photo credit: Wikipedia)

I was surprised the doubling points were not in fact sharply defined. You can see what I mean if you look closely at the first doubling point in the Wolfram Mathworld link above. Nevertheless, the doubling constant is a real constant.

English: Bifurcation diagram Česky: Bifurkační...
English: Bifurcation diagram Česky: Bifurkační diagram Polski: Zbieżność bifurkacji (Photo credit: Wikipedia)

Another sort of fractal produces tree and other diagrams that look, well, natural. A few simple rules, a few iterations and the computer draws a realistic looking skeleton tree. A few tweaks to the program and a different sort of tree is drawn. The trees are so realistic looking that it seems reasonable to conclude that there is some similarity between the underlying biological process and the underlying mathematical process. That is the biological tree is the result of an iterative process, like the mathematical trees.

Русский: Ещё одно фрактальное дерево. Фракталь...
Русский: Ещё одно фрактальное дерево. Фрактальное дерево. (Photo credit: Wikipedia)

I’ve mentioned natural objects, trees, which show fractal characteristics. Many other natural objects show such characteristics, the typical example which is usually given is that of the coastline of a country. On a large scale the coastline of a country is usually pretty convoluted, but if one zooms in the art of the coastline that one zooms in on stays pretty much as convoluted as the large scale view.

Mandelbrot fractal. Rendered as an island with...
Mandelbrot fractal. Rendered as an island with Terragen, a fractal-based landscape generator. (Photo credit: Wikipedia)

This process can be repeated right down to the point where one can see the waves. If you can imagine the waves to be frozen, then one can take the process even further, but at some point the individual water molecules become visible and the process (apparently) reaches an end.

If you want a three dimensional example, clouds, at least clouds of the same type, probably fit the bill. Basically what makes the clouds fractal is the fact that one cannot easily tell the size of a cloud if one is simple given a photograph of a cloud. It could be a huge cloud seen from a distance or a smaller cloud seen close up. Of course if one gets too close to a cloud it becomes hazy, indistinct, so one can use those clues to guess the size of a cloud.

Fractals were popularised by the mathematician Benoit Mandlebrot, who wrote about and studied the so-called Mandlebrot set, wrote about it in his book, “The Fractal Geometry of Nature”.  I’ve read this fascinating book.

English: Topological model of Mandelbrot set( ...
English: Topological model of Mandelbrot set( reflects the structure of the object ) Polski: Topologiczny model zbioru Mandelbrota ( pokazuje strukturę obiektu) (Photo credit: Wikipedia)

While I was searching for links to the Mandlebrot Set I came across the diagram which shows the correspondence of the period doubling cascade mentioned above and the Mandlebrot set. This correspondence, which I did not know about before, demonstrates the interlinked nature of fractals, and how simple mathematics can often have hidden depths. Almost always has hidden depths.

English: Paths of correspondence between scien...
English: Paths of correspondence between scientists (Photo credit: Wikipedia)